Optimal. Leaf size=55 \[ \frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e} \]
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Rubi [A] time = 0.0530198, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {1142, 14} \[ \frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e} \]
Antiderivative was successfully verified.
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Rule 1142
Rule 14
Rubi steps
\begin{align*} \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx &=\frac{f^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac{f^3 \operatorname{Subst}\left (\int \left (a x^3+b x^5+c x^7\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e}\\ \end{align*}
Mathematica [B] time = 0.0213215, size = 154, normalized size = 2.8 \[ f^3 \left (\frac{1}{4} e^3 x^4 \left (a+10 b d^2+35 c d^4\right )+\frac{1}{3} d e^2 x^3 \left (3 a+10 b d^2+21 c d^4\right )+\frac{1}{2} d^2 e x^2 \left (3 a+5 b d^2+7 c d^4\right )+d^3 x \left (a+b d^2+c d^4\right )+\frac{1}{6} e^5 x^6 \left (b+21 c d^2\right )+d e^4 x^5 \left (b+7 c d^2\right )+c d e^6 x^7+\frac{1}{8} c e^7 x^8\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 349, normalized size = 6.4 \begin{align*}{\frac{{e}^{7}{f}^{3}c{x}^{8}}{8}}+d{f}^{3}{e}^{6}c{x}^{7}+{\frac{ \left ( 15\,{d}^{2}{f}^{3}{e}^{5}c+{e}^{3}{f}^{3} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{d}^{3}{f}^{3}c{e}^{4}+3\,d{f}^{3}{e}^{2} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +{e}^{3}{f}^{3} \left ( 4\,c{d}^{3}e+2\,bde \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{4}{f}^{3}c{e}^{3}+3\,{d}^{2}{f}^{3}e \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +3\,d{f}^{3}{e}^{2} \left ( 4\,c{d}^{3}e+2\,bde \right ) +{e}^{3}{f}^{3} \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3}{f}^{3} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +3\,{d}^{2}{f}^{3}e \left ( 4\,c{d}^{3}e+2\,bde \right ) +3\,d{f}^{3}{e}^{2} \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3}{f}^{3} \left ( 4\,c{d}^{3}e+2\,bde \right ) +3\,{d}^{2}{f}^{3}e \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{2}}{2}}+{d}^{3}{f}^{3} \left ( c{d}^{4}+b{d}^{2}+a \right ) x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0143, size = 224, normalized size = 4.07 \begin{align*} \frac{1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac{1}{6} \,{\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} +{\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac{1}{4} \,{\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac{1}{3} \,{\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac{1}{2} \,{\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} +{\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45419, size = 490, normalized size = 8.91 \begin{align*} \frac{1}{8} x^{8} f^{3} e^{7} c + x^{7} f^{3} e^{6} d c + \frac{7}{2} x^{6} f^{3} e^{5} d^{2} c + 7 x^{5} f^{3} e^{4} d^{3} c + \frac{35}{4} x^{4} f^{3} e^{3} d^{4} c + \frac{1}{6} x^{6} f^{3} e^{5} b + 7 x^{3} f^{3} e^{2} d^{5} c + x^{5} f^{3} e^{4} d b + \frac{7}{2} x^{2} f^{3} e d^{6} c + \frac{5}{2} x^{4} f^{3} e^{3} d^{2} b + x f^{3} d^{7} c + \frac{10}{3} x^{3} f^{3} e^{2} d^{3} b + \frac{5}{2} x^{2} f^{3} e d^{4} b + \frac{1}{4} x^{4} f^{3} e^{3} a + x f^{3} d^{5} b + x^{3} f^{3} e^{2} d a + \frac{3}{2} x^{2} f^{3} e d^{2} a + x f^{3} d^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.100178, size = 240, normalized size = 4.36 \begin{align*} c d e^{6} f^{3} x^{7} + \frac{c e^{7} f^{3} x^{8}}{8} + x^{6} \left (\frac{b e^{5} f^{3}}{6} + \frac{7 c d^{2} e^{5} f^{3}}{2}\right ) + x^{5} \left (b d e^{4} f^{3} + 7 c d^{3} e^{4} f^{3}\right ) + x^{4} \left (\frac{a e^{3} f^{3}}{4} + \frac{5 b d^{2} e^{3} f^{3}}{2} + \frac{35 c d^{4} e^{3} f^{3}}{4}\right ) + x^{3} \left (a d e^{2} f^{3} + \frac{10 b d^{3} e^{2} f^{3}}{3} + 7 c d^{5} e^{2} f^{3}\right ) + x^{2} \left (\frac{3 a d^{2} e f^{3}}{2} + \frac{5 b d^{4} e f^{3}}{2} + \frac{7 c d^{6} e f^{3}}{2}\right ) + x \left (a d^{3} f^{3} + b d^{5} f^{3} + c d^{7} f^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09214, size = 297, normalized size = 5.4 \begin{align*} \frac{1}{8} \, c f^{3} x^{8} e^{7} + c d f^{3} x^{7} e^{6} + \frac{7}{2} \, c d^{2} f^{3} x^{6} e^{5} + 7 \, c d^{3} f^{3} x^{5} e^{4} + \frac{35}{4} \, c d^{4} f^{3} x^{4} e^{3} + 7 \, c d^{5} f^{3} x^{3} e^{2} + \frac{7}{2} \, c d^{6} f^{3} x^{2} e + c d^{7} f^{3} x + \frac{1}{6} \, b f^{3} x^{6} e^{5} + b d f^{3} x^{5} e^{4} + \frac{5}{2} \, b d^{2} f^{3} x^{4} e^{3} + \frac{10}{3} \, b d^{3} f^{3} x^{3} e^{2} + \frac{5}{2} \, b d^{4} f^{3} x^{2} e + b d^{5} f^{3} x + \frac{1}{4} \, a f^{3} x^{4} e^{3} + a d f^{3} x^{3} e^{2} + \frac{3}{2} \, a d^{2} f^{3} x^{2} e + a d^{3} f^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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